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An underwater localization scheme for sparse sensing acoustic positioning in stratified and perturbed UASNs

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Abstract

There is a dearth of compressed-sensing localization in the domain of underwater acoustic sensor networks (UASNs). To address this limitation, this paper proposes SAP2: a Sparse-sensing Acoustic Positioning scheme for Stratified and Perturbed UASNs. Three conditions of underwater perturbation are defined, namely, Spin perturbation, Rectilinear dispersion condition, and Radially perturbed condition to model the sudden disturbances in the underwater network topology. Then, “Footprint Variation” is proposed to express the behavior of UASN node topology. Theorem 2.2.b is proposed for the exact retrieval of sparsely deployed UASN location under perturbation conditions. The derivation of a unique solution to the insertion vector in Lemma 3.1.a holds the estimated location of target, necessitating the incoherence of the sensing matrix with the basis vector. Block matrix representation of the coefficients is derived in Lemma 3.1.b. Invertibility of the block matrix is proven in Lemma 3.1.c to establish sparsity of underwater sensor networks. An example is illustrated with a realistic scenario to observe parameters such as correlation of sparse coefficients, distinctness of discretized samples and footprint variation based on perturbation. Footprint variation is found to be below 100m2 for uncorrelated discrete measurements using SAP2 technique. Through rigorous mathematical justification and realistic underwater channel scenario, it is established that compressed sensing based underwater localization is feasible for a sparse sensor network.

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Abbreviations

\({\mathbf{A}}\left( \alpha \right)\) :

Function representing fractional loss in grid insertion

\(a\), \(b\) :

Distinct solution to discrete UASN positioning equations

\(C^{A.S}\), \(C^{{\left( {n.A.S} \right)}}\) :

Positive constants for acoustically stratified and unstratified path, respectively

\(C_{g}\), \(C_{g}^{{\left( {A.S} \right)}}\), \(C_{g}^{{\left( {n.A.S} \right)}}\) :

Cumulative covariances: overall, acoustically stratified, acoustically unstratified

\({\mathbf{d}}\) :

Target baseline measurements

\(d\left( {\alpha_{i} ,\alpha_{j} } \right)\) :

Generalized distance between ith and jth instance of triangulation \(\alpha\)

\({\mathbf{F}}\) :

Sound speed profile vector

\(g\) :

Tidal constant

\(G\) :

Discrete lattice framework

\(h_{n,m}\) :

Discrete time media interface between \(m^{th}\) UAN and \(n^{th}\) USN

\(K\) :

Function on which grid insertion depends

\(L.L.R^{p} \left( {x_{m} \left( j \right)} \right)\) :

Log likelihood ratio of \(x_{m} \left( j \right)\)

\({\mathbf{m}}\) :

Orthogonal coefficient vector

\(\left( {n.A.S} \right)\),\(\left( {A.S} \right)\) :

Acoustically unstratified path, acoustically stratified path

\({\mathbf{s}}\) :

Sensing matrix

\({\mathbf{t}}_{x}\) :

Number of transmission round trip times

\(V\left( \alpha \right)\) :

Insertion function for triangulation \(\alpha\)

\(\delta_{n,m}^{\left( l \right)} \left( j \right)\) :

Transient ocean current perturbation

\(\mu\) :

Perturbation sensitivity matrix

\(\rho_{\alpha }\) :

Correlation function

\(\Theta\) :

Vector which stores indices of non-zero elements

\(\Delta\) :

Generalized Footprint variation

\(\Phi\) :

Search space for target in the next discrete-time domain

\(\varepsilon\) :

Measurement noise

\(\hat{\wp }\left( \alpha \right)\) :

Time shift within the insertion function

AUV:

Autonomous underwater vehicle

CRLB:

Cramer Rao lower bound

UAN:

Underwater anchor nodes

UASN:

Underwater Acoustic Sensor network

USN:

Underwater sensor nodes

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Acknowledgements

This research was supported by the Ministry of Electronics and Information Technology, Govt. of India

Funding

Ministry of Electronics and Information technology,13(29)/2020-CC&BT,Rajeev Arya.

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  1. Wireless Sensor Networks Lab, Department of Electronics and Communication Engineering, National Institute of Technology, Patna, Bihar, 800005, India

    Prateek & Rajeev Arya

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Prateek, Arya, R. An underwater localization scheme for sparse sensing acoustic positioning in stratified and perturbed UASNs. Wireless Netw 28, 241–256 (2022). https://doi.org/10.1007/s11276-021-02839-0

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